DSpace at VNU: On the radical characteristic of regularities

VNU JOURNAL OF SCIENCE, Mathematics - Physics, t XVIII n°l - 2002 ON T H E RA DICA L C H A R A C T E R IS T IC OF R E G U L A R IT IE S T n T r o n g H u e Faculty o f M athem atics College o f N atural Sciences, Vietnam National University Hanoi I I n t r o d u c t io n In this paper we shall work in the variety w o f algebra s over an associative and com m utative ring A which unity elem ent For a given subclass I Ỉ of th e variety 11 a /K1 ]f (») « x < - B S i t whore = ( / / ) s - regular alợrhra An elem ent a of the algebra is called 5-regular if a G - s , \ An algebra is said to 1)0 S-regulai if ẦS \ = A All ideal / of an algebra A is called - regular if I is an 5-rogular algebra P r o p e r t i e s We obviously have the following P r o p o s it io n I f tin class s IS a regularity then S /t ( ( 0 , ) ) = f o r every algebra A of the class IF P r o p o s it io n The class of the S-re.qular algebras is hom om orphically closed Proof Let B be ail image o f an 5-regular algebra A under the hom omorphism / Now let b be an arbitrary elem ent of B Then there exists an elem ent a of A such that l> = / ( a ) Sin ce A is an 5-regular algebra there is an elem ent X o f A such that s A ( x ) ~ a B y th e com m utative diagram (a) we have: h = f(a) = f ( S A(;/•)) = S Bl f * ( x ) ) * S ) - B - Therefore' the algebra B is S-rogular T h e o r e m I f the class s — { s \ : A ^ —> w4}.tei\ is a regularity then the class of all S-rcỊỊỉUar algebra IS a radical class in II’ i f and only i f the following condition is satisfied If I is an S -rcyular ideal of the algebra A and f o r every element a of A there exists an (devient Vo f A * such that S a (j:) - a = m od I , then A is a regular algebra Proof A ssum e that the class R of all 5-regular algebras is a radical class Now suppose that / is an 5-regular ideal o f an algebra and for every a of A there exists an element X of A'*' such that S a ( x ) — a = m od / W e have to show that the algebra -4 is S-regular Let us consider th e fact or algebra A / Take any elem ent ã of A ị I By hypothesis there ex ists all elem ent X of A 00 such th at S - A { x ) - a = mod / So in Tran 'Trong H u e \ the factor algebra A / I the equality (I = S a ( x ) holds /) : /1 * A / I we have the com m utative diagram For the natural hom om orphism A 00 p (b ) A /I We have (1 = S a ( x ) = p(S (x )) = ^ //( p ° ° ( x ) ) Therefore th e elem ent Ỡ is S regular T his implies the 5-regularit.v of th e algebra A / I Since radical classes are closed under extensions, th e algebra A is 5-regular Conversely, assum e th at the 5-regularity satisfies the condition o f th e theorem We shall show that the class R of all 5-regular algebras is a radical class Clearly, the class R is not empty By proposition th e class R is honiom orphically closed T h e d ition (i) o f th e radical property is satisfied Now suppose that for an ideal of an algebra A, both / and A / I are /?-algebra Since the algebra A / I is 9-regular therefore for every elem ent a o f A there exists an element of (A /I)'* - such that S a / i { x ) = Ã By th e com m utative diagram (b) we have a = ^ //( x ) = S A/i(p°°{x) ) = p(S>i(x)) = Srf(x) This implies s.,t(;r) - a s mod / By the condition of theorem th e algebra /1 is 5-regular Hence tho class R is closed under extensions T he condition (iii) o f th e radical property is satisfied By the proposit ion th e zero ideal o f an algebra A is an 5-regular ideal Hence the set of the R -ideals o f algebra A is not em pty Su ppose both I\ and /*2 be R -ideals of the algebra ,4 By th e second isom orphism theorem we have h ± h / h 11 n /2 Since the class R is hom om orphicallv closed and closed under exten sion s, th e above iso­ morphism im plies th at /1 + 12 is an /ĩ-algeb By a sim ple induction we can prove that the sum of any finite number o f I Ỉ - ideals o f th e algebra A is again an /?-ideal Filially, we have to show that the sum R ( A ) o f ail fỵ-ideals o f the algebra A is an 5-regular ideal Take any elem ent a of R (A ) T hen there are thí' R n- ideals / such th at th e ideal J = I h- contains the elem ent a Sine J is an S-regular ideal there is an elem ent JT of J x such th at s /( x ) = a For th e em bedding i j : J —> /?(v4 ) we have th e com m utative diagram O f / Therefor th e ideal /Ỉ(A ) is 5-regular T his com pletes th e proof o f th e theorem As the radical criterions o f 5-regularities we have the following assertions (c) n the radical chavuctf'.ristic o f regularities P r o p o s i t i o n The class of all S -rcỊỊu lar (dọcbras is a radical, class if the following con­ dition is satisfied F o r arbitrary elements (Ï ill an alqebra A, and X of A™ i f the element S \(:v) - a is s ' I f (Ịiỉlíỉ I then iln ỊÉlcmrtì1 a is also s -vcọulav Proof \Y A}.4ỗvr0 * .(12 )) = A ll elem ent a o f an algebra A is said to be regular in the souse o f Neumann [14| if a € (lAa Clearly 1-regular coincides w ith the regularity ill the sense of Neum ann 6^(( B.Brown and N.H Mccov T h e radical o f a ring D uke Math 499 S oc (1955) 511 15(1948) 495 - G B Brown and N.H Mccoy T he m axim al regular ideal o f a ring, Proc Amer Math Soc 1(1950), 165 - 171 N Divinsky Pseudo-regularity, Canad J Math., (1958), 401 - 410 Oil the radical characteristic o f regularities r N I L Colliding ami A II Ortiz Structure of st'tnipriim’ (]) q) - radicals 10 11 12 i:i I 15 l(i 17 Is 19 I ’at ./ Math 37(1 ), 97 - 9!) T T Mue and F S /a s / Oil the radical classes (k'tmniiiLHl by regularities Acta Sri M ad) ( 1!)M ) i :h - ITO N .Jacobson Some remarks on one-sided inverses, Pvoc Amur Math Sot' 1(1950) 352 - :r>:> A (Ỉ Kurosli Radicals of rings and algebras Mat SI)., 3 (7 ) (1953) 13 20 .) D Kcknight and G L M usser Special (p.q) - radicals Canad •/ Math (1 ) - 44 c L M ussel- Linear scm iprim e (p.q) -radical Pac ./ Math (1 ) 749 - 757 Von N eum ann On regular rings Proc Math Acad Sri 2 (1 ) 7 - 783 S Perl is A charactrm aU on o f th e radical o f an algebra Bull Am er Math Sor., 48(11)42) 128 - 132 c R ous lieyuluritic.s o f rings D issertation D elft 1975 13 D(' La R ose U rals and radicals D issertation Delft 1970 F A Szasz Radiacals of lings Akadm iai Kind Budapest 1981 H Wiô'garnit Radical and sem isiniple classes o f rings Quern's Papers III Pirn (111(1 A/ipl M oth V o l ( K in g s t o n ) TAP C.HI KHOA HOC DHQGHN Tốn • Ly t XVIII n°l - 2002 VỀ Đ Ặ C TRUNc; C A N C Ủ A CÁ C TÍNH CHAT C H ÍN H Q U Y Trần Trong Huệ Kltoa Toán - C - Till học D ili học K h o a học T ự Iiliiên - D H Q C Ì H ù N ộ i XĨI \ v hì đa lạp dại sỏ (khơng thiết kêì hợp) trẻn vành K giao hốn có đ n v ị V i m ỗ i đ i s ỏ A th u ộ c i r k ý h iệ u A * t ổ n g trự c t iế p A „ p l u.sA.ij M ỗ i lớ p ánh \ s = {5.4 : 4X > li gọi tính chất s - quy điều kiện sau dược thoá mãn: Đùi với A , B thuộc IV / thuộc h o n i K ( A R ) ta có hộ thức giao hoán f s ( - $ i t f x / x = ( / / )• Phần tứ n cùa đại SC) gọi S- quy nêu (I ị- Ò.S' ( Đại s ố A aọi -ch ín h quy nêu 'iS\\ — A Iđêiiii Ị cua đại số A gọi lit Scliính quy đại số S- quy Trong báo chúng tỏi chứng minh ràng tính chất s - quy I11 ỘI l í n h c h t c n t h e o n g h ĩ a K u r o s h v A m i t s u r k h i v c h i k h i đ i ổ u k i ệ n s a u đ ợ c t h o i i m ãn: Nếu / iđêan s - quy đại sơ A a € A tổn phần tử /• f chos.\(.r) - a = mod I A đại số S - quy Từ đặc trưng ta chứng minh dược hai điéu kiện đủ đổ tính chất s - quy t í n h c h ấ t c ă n Trong trườn” hợp 11' đa tạp đại sổ kết hợp khái niệm k é t q u c ú a b ib o n y S- quy s ự t ổ n g q u t h o c c t í n h c h t c h í n h q u y c ù a c c tá c g iá Von Neum ann, Perlis (cán Jacobson) Brown - McCoy, Kaplansky, De la R ose, Divinsky, Blair M aknish - Musser, v.v ... rse regularities are the radical properties in the srnsc of Kurosli